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A154343 S(n,k) an additive decomposition of the Springer number (generalized Euler number), (triangle read by rows). 6
1, 3, -2, 9, -16, 4, 27, -98, 60, 0, 81, -544, 616, 0, -96, 243, -2882, 5400, 0, -3360, 960, 729, -14896, 43564, 0, -72480, 46080, -5760, 2187, -75938, 334740, 0, -1246560, 1323840, -362880, 0, 6561, -384064, 2495056, 0, -18801216, 29675520 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The Swiss-Knife polynomials A153641 can be understood as a sum of polynomials. Evaluated at x=1/2 and multiplied by 2^n these polynomials result in a decomposition of the Springer numbers A001586.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1274

Peter Luschny, The Swiss-Knife polynomials.

FORMULA

Let c(k) = frac{(-1)^{floor(k/4)}{2^{floor(k/2)}} [4 not div k] (Iverson notation).

S(n,k) = Sum_{v=0,..,k} ( (-1)^(v)*binomial(k,v)*2^n*c(k)*(v+3/2)^n );

S(n) = Sum_{k=0,..,n} S(n,k).

EXAMPLE

1,

3,     -2,

9,     -16,      4,

27,    -98,     60,      0,

81,    -544,    616,     0,   -96,

243,   -2882,   5400,    0,  -3360,      960,

729,   -14896,  43564,   0,  -72480,    46080,     -5760,

2187,  -75938,  334740,  0, -1246560,  1323840,   -362880,  0,

6561, -384064,  2495056, 0, -18801216, 29675520, -13386240, 0, 645120.

MAPLE

S := proc(n, k) local v, c; c := m -> if irem(m+1, 4) = 0 then 0 else 1/((-1)^iquo(m+1, 4)*2^iquo(m, 2)) fi; add((-1)^(v)*binomial(k, v)*2^n*c(k)*(v+3/2)^n, v=0..k) end: seq(print(seq(S(n, k), k=0..n)), n=0..8);

MATHEMATICA

c[m_] := If[Mod[m+1, 4] == 0, 0, 1/((-1)^Quotient[m+1, 4]*2^Quotient[m, 2])]; s[n_, k_] := Sum[(-1)^v*Binomial[k, v]*2^n*c[k]*(v+3/2)^n, {v, 0, k}]; Table[s[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jul 30 2013, after Maple *)

CROSSREFS

Cf. A153641, A154341, A154342, A154344, A154345.

Sequence in context: A053088 A077898 A076584 * A049969 A088634 A118791

Adjacent sequences:  A154340 A154341 A154342 * A154344 A154345 A154346

KEYWORD

easy,sign,tabl

AUTHOR

Peter Luschny, Jan 07 2009

STATUS

approved

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Last modified September 22 09:43 EDT 2017. Contains 292337 sequences.